{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:YD5RAJWMV2OYIOX4WER6LMIRKZ","short_pith_number":"pith:YD5RAJWM","canonical_record":{"source":{"id":"1009.2861","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-15T08:27:14Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c8849b63e85ecd9c47cdb7aa5e60a17be77696d04d9ffbdf9c3d383ece7dfad6","abstract_canon_sha256":"2ff1e76033eaca251918a6b74d51e4891da3e2e8ae32bee865524596d0b9faf1"},"schema_version":"1.0"},"canonical_sha256":"c0fb1026ccae9d843afcb123e5b111565caf8bab05b32357d546ed2d021a8681","source":{"kind":"arxiv","id":"1009.2861","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2861","created_at":"2026-05-18T03:48:34Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2861v3","created_at":"2026-05-18T03:48:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2861","created_at":"2026-05-18T03:48:34Z"},{"alias_kind":"pith_short_12","alias_value":"YD5RAJWMV2OY","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YD5RAJWMV2OYIOX4","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YD5RAJWM","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:YD5RAJWMV2OYIOX4WER6LMIRKZ","target":"record","payload":{"canonical_record":{"source":{"id":"1009.2861","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-15T08:27:14Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c8849b63e85ecd9c47cdb7aa5e60a17be77696d04d9ffbdf9c3d383ece7dfad6","abstract_canon_sha256":"2ff1e76033eaca251918a6b74d51e4891da3e2e8ae32bee865524596d0b9faf1"},"schema_version":"1.0"},"canonical_sha256":"c0fb1026ccae9d843afcb123e5b111565caf8bab05b32357d546ed2d021a8681","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:34.455830Z","signature_b64":"fcnjR4/gQMA4CSu+FsqGmnaniREP6L3otavmqmWowNnoK9ObL7XViH2zdn8U8GwHiCFd4W1MeoHeqvExaiK5CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0fb1026ccae9d843afcb123e5b111565caf8bab05b32357d546ed2d021a8681","last_reissued_at":"2026-05-18T03:48:34.455109Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:34.455109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.2861","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:48:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CP1YbXAm1pg8bks+RaGZmn490eWpcatTxK940k/SPB0T7OoPQg57N7jwZ8NR4IZD89+7xXRXLHBLlVCKgPGqDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:14:24.582306Z"},"content_sha256":"592c4d31d1ed1fadbef9c566074384cc1624c5d923221f2b6d375c3be3eea67f","schema_version":"1.0","event_id":"sha256:592c4d31d1ed1fadbef9c566074384cc1624c5d923221f2b6d375c3be3eea67f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:YD5RAJWMV2OYIOX4WER6LMIRKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Edge-intersection graphs of grid paths: the bend-number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Daniel Heldt, Kolja Knauer, Torsten Ueckerdt","submitted_at":"2010-09-15T08:27:14Z","abstract_excerpt":"We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum $k$ such that grid-paths with at most $k$ bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every $k$ there is a graph with bend-number $k$. Moreover we provide new upper and lower bounds of the bend-number of graphs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2861","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:48:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"if3PQ5qrkvnuZ0u0FOMd/scpPxmevlrMQg5X6vbuKGd0bXRfKJCyhpgnH3tmqMJSZl3wJuI+ht3CnZgbr+zBCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:14:24.582740Z"},"content_sha256":"d8ce9d2246b1e6e257548763d23ea6339e7c7284918134198bcb3b88d6d7fd22","schema_version":"1.0","event_id":"sha256:d8ce9d2246b1e6e257548763d23ea6339e7c7284918134198bcb3b88d6d7fd22"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ/bundle.json","state_url":"https://pith.science/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T19:14:24Z","links":{"resolver":"https://pith.science/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ","bundle":"https://pith.science/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ/bundle.json","state":"https://pith.science/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YD5RAJWMV2OYIOX4WER6LMIRKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YD5RAJWMV2OYIOX4WER6LMIRKZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2ff1e76033eaca251918a6b74d51e4891da3e2e8ae32bee865524596d0b9faf1","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-15T08:27:14Z","title_canon_sha256":"c8849b63e85ecd9c47cdb7aa5e60a17be77696d04d9ffbdf9c3d383ece7dfad6"},"schema_version":"1.0","source":{"id":"1009.2861","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2861","created_at":"2026-05-18T03:48:34Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2861v3","created_at":"2026-05-18T03:48:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2861","created_at":"2026-05-18T03:48:34Z"},{"alias_kind":"pith_short_12","alias_value":"YD5RAJWMV2OY","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YD5RAJWMV2OYIOX4","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YD5RAJWM","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:d8ce9d2246b1e6e257548763d23ea6339e7c7284918134198bcb3b88d6d7fd22","target":"graph","created_at":"2026-05-18T03:48:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum $k$ such that grid-paths with at most $k$ bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every $k$ there is a graph with bend-number $k$. Moreover we provide new upper and lower bounds of the bend-number of graphs","authors_text":"Daniel Heldt, Kolja Knauer, Torsten Ueckerdt","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-15T08:27:14Z","title":"Edge-intersection graphs of grid paths: the bend-number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2861","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:592c4d31d1ed1fadbef9c566074384cc1624c5d923221f2b6d375c3be3eea67f","target":"record","created_at":"2026-05-18T03:48:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2ff1e76033eaca251918a6b74d51e4891da3e2e8ae32bee865524596d0b9faf1","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-15T08:27:14Z","title_canon_sha256":"c8849b63e85ecd9c47cdb7aa5e60a17be77696d04d9ffbdf9c3d383ece7dfad6"},"schema_version":"1.0","source":{"id":"1009.2861","kind":"arxiv","version":3}},"canonical_sha256":"c0fb1026ccae9d843afcb123e5b111565caf8bab05b32357d546ed2d021a8681","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0fb1026ccae9d843afcb123e5b111565caf8bab05b32357d546ed2d021a8681","first_computed_at":"2026-05-18T03:48:34.455109Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:34.455109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fcnjR4/gQMA4CSu+FsqGmnaniREP6L3otavmqmWowNnoK9ObL7XViH2zdn8U8GwHiCFd4W1MeoHeqvExaiK5CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:34.455830Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2861","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:592c4d31d1ed1fadbef9c566074384cc1624c5d923221f2b6d375c3be3eea67f","sha256:d8ce9d2246b1e6e257548763d23ea6339e7c7284918134198bcb3b88d6d7fd22"],"state_sha256":"08e43cfe273b3847159df4742e190b19bad1b27304d452c43f0a349a2f618d62"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ARAerphXk4h54dAFxcV/ZWJxBZXh+bS3A36NM3pHJ4MVHMIbIY36yoVK7xZTJ/xDzi1FEVaikBqXHyB5s5e5Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T19:14:24.585353Z","bundle_sha256":"58b84771c57b7e25f162d67e580d195f3094610e4371b474a58474e33c8ebe5e"}}